A Knowledge-Sensitive On-line Exercise for Developing Algebraic Calculation Strategies

Abstract

Our observations indicated that students did not learn the calculation of algebraic rational expressions easily even if the students had enough skills to calculate numerical fractions. We suspected the reason is that algebraic calculations depend on several mathematical rewriting rules, which are usually hidden in black boxes at numerical calculations, and the students can not use the rules properly. The students can calculate numerical fractions automatically according to a few patterns they have learnt from experience and without thinking of the mechanism of the calculating steps. They can calculate even polynomials in such a way. They are surprised to find that their pattern-matching algorithms often fail in the calculation of rational expressions. Their patterns sometimes lead them to typical calculating errors lead by incorrect calculating rules, but they can not separate such kind of serious errors from other errors caused by careless manipulation.

Teaching rule-based calculation in a class was not enough for these students. They thought that the rules were too obvious to learn and they knew the rules already. But knowing the rules and using them properly are separate things. Traditional exercises of calculations improved the students' manipulating skills and mostly decreased the errors caused by carelessness, but they did not improve the students' understanding on mathematical rewriting rules very much. The students, who could not separate the errors caused by incorrect rules from careless mistakes, were simply satisfied by the fewer but still existing ratio of the errors. They were repeating the errors lead by incorrect rewriting rules.

In this paper, the authors propose a new on-line exercise for such students to learn the calculation of algebraic rational expressions. In the exercise, a series of calculating steps from the question to the answer are displayed on a student's screen. Some of the steps may include calculating errors, and the others do not. The student is asked to examine every calculating step, determine the type of the error for an erroneous step, and describe the rewriting rule of a correct step.

Such an exercise, when it was done on a pencil-and-paper basis, measured the students' sensitivities to calculating errors and their affinities to rule-based calculation.

The new exercise, if it is well organized with a manipulating exercise and an interactive instruction, is expected to help the students, who do not recognize the importance of mathematical rules in calculations, to develop the strategies themselves. We expect the system to help the students who used to improve very slowly in algebraic calculations even after a series of traditional calculating exercises.